Hence the curvature corrections become only a shift to an effective mass and are effectively erased from the EOM's. But unfortunately, this does not hold for arbitrary \(f(z)\)  where the derivative of the metric \(f^{\prime }(z)\)  is not proportional to \(f(z)\) , spoiling the convenient relation from AdS.
Meanwhile, the coupling to the gauge field \(\phi \)  also complicates the argumentation and sabotages some ideas.
In fact, one very promising path to present the EOM's in a neat form leads down to exploiting the invariance of an action of the form Eq. (REF ) under gauge transformations \(\delta B\)  for free fields in AdS-like theories, as shown for instance in Ref. [1]}. In that paper it is shown that by electing an appropriate gauge, the fields can be rescaled in such a way that the action only depends on partial derivatives of these fields and the EOM's can directly be read off. But even though this method could in general take care of general metrics by adapting the field rescaling, it is easy to see that it eventually breaks down when including gauge field interactions via gauge-covariant derivatives.
